Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
stochastic processes; measurable modification
stochastic processes; measurable modification
Summary:
We present, with purely didactic aims, a simple and essentially self-contained proof of two necessary and sufficient conditions for existence of a measurable modification of a stochastic process with values in a separable complete metric space. Existence of a measurable modification of a stochastic process continuous in probability is an immediate consequence.
References:
[1] Burke, M. R., Macheras, N. D., Strauss, W.: Measurability of stochastic processes on topological probability spaces. Topology Appl. 370 (2025), 109438, 37 pp. DOI
[2] Cohn, D. L.: Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 161-165. DOI
[3] Chung, K. L., Doob, J. L.: Fields, optionality and measurability. Amer. J. Math. 87 (1965), 397-424. DOI 10.2307/2373011
[4] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge 1992.
[5] Dellacherie, C., Meyer, P.-A.: Probabilités et potentiel, Chapitres I à IV. Hermann, Paris 1975.
[6] Doob, J. L.: Stochastic Processes. Wiley, New York 1953.
[7] Fremlin, D. H.: Measure Theory, Vol. 2. Second edition. Torres Fremlin, Colchester 2010.
[8] Geiss, H., Geiss, S.: Measure, Probability and Functional Analysis. Springer, Cham 2025.
[9] Gikhman, I. I., Skorokhod, A. V.: Introduction to the Theory of Random Processes. W. B. Saunders Co., Philadelphia 1969.
[10] Gihman, I. I., Skorohod, A. V.: The Theory of Stochastistic Processes I. Springer-Verlag, New York 1974.
[11] He, S.-w., Wang, J.-g., Yan, J.-a.: Semimartingale Theory and Stochastic Calculus. CRC, Boca Raton 1992.
[12] Hoffmann-Jørgensen, J.: Existence of measurable modifications of stochastic processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 205-207. DOI
[13] Hoffmann-Jørgensen, J.: Stochastic Processes on Polish Spaces. Various Publ. Ser. (Aarhus) 39, Aarhus Universitet, Matematisk Institut, Aarhus 1991.
[14] Kawada, Y.: On the measurable stochastic processes. Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 512-527.
[15] Klenke, A.: Probability Theory. Third edition. Springer, Cham 2020.
[16] Mansfield, R., Weitkamp, S. G.: Recursive Aspects of Descriptive Set Theory. Oxford University Press, New York 1985.
[17] Meyer, P.-A.: Probability and Potentials. Blaisdell, Waltham 1966.
[18] Neveu, J.: Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco 1965.
[19] Ondreját, M., Seidler, J.: On existence of progressively measurable modifications. Electron. Commun. Probab. 18 (2013), 20, 6 pp. DOI
[20] Schilling, R. L., Kühn, F.: Counterexamples in Measure and Integration. Cambridge University Press, Cambridge 2021. DOI
[21] Skorokhod, A. V.: On measurability of stochastic processes. Theory Probab. Appl. 25 (1980), 139-141. DOI
[22] Srivastava, S. M.: A Course on Borel Sets. Springer, New York 1998.
Search
Partner of
